There is a layer of snow $x \ cm$ thick on water,when the temperature of the air is $-\theta ^\circ C$ (less than the freezing point). If the thickness of the layer increases from $x$ to $y$ in time $t$,then the value of $t$ is given by:

Three rods made of the same material and having the same cross-sectional area but different lengths $10\, cm, 20\, cm$,and $30\, cm$ are joined as shown. The temperature of the junction is ......... $^oC$.

Two ends of rods of length $L$ and radius $r$ of the same material are kept at the same temperature difference. Which of the following rods conducts the most heat?

Two walls of thicknesses $d_1$ and $d_2$ and thermal conductivities $k_1$ and $k_2$ are in contact. In the steady state,if the temperatures at the outer surfaces are $T_1$ and $T_2$,the temperature at the common wall is

$A$ large cylindrical rod of length $L$ is made by joining two identical rods of copper and steel of length $(\frac{L}{2})$ each. The rods are completely insulated from the surroundings. If the free end of the copper rod is maintained at $100\,^oC$ and that of the steel rod at $0\,^oC$,then the temperature of the junction is........$^oC$ (Thermal conductivity of copper is $9$ times that of steel).

The temperature difference across two cylindrical rods $A$ and $B$ of the same material and same mass are $40^{\circ} C$ and $60^{\circ} C$ respectively. In steady state,if the rates of flow of heat through the rods $A$ and $B$ are in the ratio $3: 8$,the ratio of the lengths of the rods $A$ and $B$ is (in $: 3$)